Superconducting resonator definition based on one or more attributes of a superconducting circuit

ABSTRACT

Systems, computer-implemented methods, and computer program products that can facilitate superconducting resonator definition based on one or more superconducting circuit attributes, are described. According to an embodiment, a system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can comprise a resonant circuit component that derives a resonant circuit indicative of a superconducting resonator of a superconducting circuit based on one or more attributes of the superconducting circuit. The computer executable components can further comprise a resonator definition component that defines a frequency value of the superconducting resonator based on the resonant circuit.

BACKGROUND

The subject disclosure relates to superconducting resonators, and morespecifically, to superconducting resonator definition based on one ormore superconducting circuit attributes.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements, or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, devices, systems, computer-implemented methods, and/orcomputer program products that can facilitate superconducting resonatordefinition based on one or more superconducting circuit attributes aredescribed.

According to an embodiment, a system can comprise a memory that storescomputer executable components and a processor that executes thecomputer executable components stored in the memory. The computerexecutable components can comprise a resonant circuit component thatderives a resonant circuit indicative of a superconducting resonator ofa superconducting circuit based on one or more attributes of thesuperconducting circuit. The computer executable components can furthercomprise a resonator definition component that defines a frequency valueof the superconducting resonator based on the resonant circuit.

According to another embodiment, a computer-implemented method cancomprise deriving, by a system operatively coupled to a processor, aresonant circuit indicative of a superconducting resonator of asuperconducting circuit based on one or more attributes of thesuperconducting circuit. The computer-implemented method can furthercomprise defining, by the system, a frequency value of thesuperconducting resonator based on the resonant circuit.

According to another embodiment, a computer program product that canfacilitate superconducting resonator definition based on one or moresuperconducting circuit attributes. The computer program product cancomprise a computer readable storage medium having program instructionsembodied therewith, the program instructions can be executable by aprocessor to cause the processor to derive, by the processor, a resonantcircuit indicative of a superconducting resonator of a superconductingcircuit based on one or more attributes of the superconducting circuit.The program instructions can be further executable by the processor tocause the processor to define, by the processor, a frequency value ofthe superconducting resonator based on the resonant circuit.

According to an embodiment, a system can comprise a memory that storescomputer executable components and a processor that executes thecomputer executable components stored in the memory. The computerexecutable components can comprise a circuit effect component thatdetermines a kinetic inductance value of a superconducting circuit. Thecomputer executable components can further comprise a resonatordefinition component that defines one or more parameters of asuperconducting resonator based on the kinetic inductance value.

According to another embodiment, a computer-implemented method cancomprise determining, by a system operatively coupled to a processor, akinetic inductance value of a superconducting circuit. Thecomputer-implemented method can further comprise defining, by thesystem, one or more parameters of a superconducting resonator based onthe kinetic inductance value.

DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of an example, non-limiting systemthat can facilitate superconducting resonator definition based on one ormore superconducting circuit attributes in accordance with one or moreembodiments described herein.

FIG. 2 illustrates a block diagram of an example, non-limiting systemthat can facilitate superconducting resonator definition based on one ormore superconducting circuit attributes in accordance with one or moreembodiments described herein.

FIGS. 3A and 3B illustrate example, non-limiting circuits that canfacilitate superconducting resonator definition based on one or moresuperconducting circuit attributes in accordance with one or moreembodiments described herein.

FIG. 4 illustrates an example, non-limiting circuit that can facilitatesuperconducting resonator definition based on one or moresuperconducting circuit attributes in accordance with one or moreembodiments described herein.

FIG. 5 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.

FIG. 6 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.

FIG. 7 illustrates an example, non-limiting table that can facilitatesuperconducting resonator definition based on one or moresuperconducting circuit attributes in accordance with one or moreembodiments described herein.

FIG. 8 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.

FIG. 9 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.

FIG. 10 illustrates a block diagram of an example, non-limitingoperating environment in which one or more embodiments described hereincan be facilitated.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details. It is noted that thedrawings of the present application are provided for illustrativepurposes only and, as such, the drawings are not drawn to scale.

Quantum computing is generally the use of quantum-mechanical phenomenafor the purpose of performing computing and information processingfunctions. Quantum computing can be viewed in contrast to classicalcomputing, which generally operates on binary values with transistors.That is, while classical computers can operate on bit values that areeither 0 or 1, quantum computers operate on quantum bits that comprisesuperpositions of both 0 and 1, can entangle multiple quantum bits(qubits), and use interference.

Universal Quantum computers require a large number of physical qubits(e.g., millions), and for each qubit, one or more readout resonatorsand/or bus resonators are needed. Resonators are implemented usingtransmission lines. Time and effort to design and/or simulatesuperconducting resonators with proper accuracy grows rapidly.

FIG. 1 illustrates a block diagram of an example, non-limiting system100 that can facilitate superconducting resonator definition based onone or more superconducting circuit attributes in accordance with one ormore embodiments described herein. According to several embodiments,system 100 can comprise a resonator definition system 102. In someembodiments, resonator definition system 102 can comprise a memory 104,a processor 106, a resonant circuit component 108, a resonatordefinition component 110, and/or a bus 112.

It should be appreciated that the embodiments of the subject disclosuredepicted in various figures disclosed herein are for illustration only,and as such, the architecture of such embodiments are not limited to thesystems, devices, and/or components depicted therein. For example, insome embodiments, system 100 and/or resonator definition system 102 canfurther comprise various computer and/or computing-based elementsdescribed herein with reference to operating environment 1000 and FIG.10. In several embodiments, such computer and/or computing-basedelements can be used in connection with implementing one or more of thesystems, devices, components, and/or computer-implemented operationsshown and described in connection with FIG. 1 or other figures disclosedherein.

According to multiple embodiments, memory 104 can store one or morecomputer and/or machine readable, writable, and/or executable componentsand/or instructions that, when executed by processor 106, can facilitateperformance of operations defined by the executable component(s) and/orinstruction(s). For example, memory 104 can store computer and/ormachine readable, writable, and/or executable components and/orinstructions that, when executed by processor 106, can facilitateexecution of the various functions described herein relating toresonator definition system 102, resonant circuit component 108,resonator definition component 110, and/or another component associatedwith system 100 and/or resonator definition system 102, as describedherein with or without reference to the various figures of the subjectdisclosure.

In some embodiments, memory 104 can comprise volatile memory (e.g.,random access memory (RAM), static RAM (SRAM), dynamic RAM (DRAM), etc.)and/or non-volatile memory (e.g., read only memory (ROM), programmableROM (PROM), electrically programmable ROM (EPROM), electrically erasableprogrammable ROM (EEPROM), etc.) that can employ one or more memoryarchitectures. Further examples of memory 104 are described below withreference to system memory 1016 and FIG. 10. Such examples of memory 104can be employed to implement any embodiments of the subject disclosure.

According to multiple embodiments, processor 106 can comprise one ormore types of processors and/or electronic circuitry that can implementone or more computer and/or machine readable, writable, and/orexecutable components and/or instructions that can be stored on memory104. For example, processor 106 can perform various operations that canbe specified by such computer and/or machine readable, writable, and/orexecutable components and/or instructions including, but not limited to,logic, control, input/output (I/O), arithmetic, and/or the like. In someembodiments, processor 106 can comprise one or more central processingunit, multi-core processor, microprocessor, dual microprocessors,microcontroller, System on a Chip (SOC), array processor, vectorprocessor, and/or another type of processor. Further examples ofprocessor 106 are described below with reference to processing unit 1014and FIG. 10. Such examples of processor 106 can be employed to implementany embodiments of the subject disclosure.

In some embodiments, resonator definition system 102, memory 104,processor 106, resonant circuit component 108, resonator definitioncomponent 110, and/or another component of resonator definition system102 as described herein can be communicatively, electrically, and/oroperatively coupled to one another via a bus 112 to perform functions ofsystem 100, resonator definition system 102, and/or any componentscoupled therewith. In several embodiments, bus 112 can comprise one ormore memory bus, memory controller, peripheral bus, external bus, localbus, and/or another type of bus that can employ various busarchitectures. Further examples of bus 112 are described below withreference to system bus 1018 and FIG. 10. Such examples of bus 112 canbe employed to implement any embodiments of the subject disclosure.

In some embodiments, resonator definition system 102 can comprise anytype of component, machine, device, facility, apparatus, and/orinstrument that comprises a processor and/or can be capable of effectiveand/or operative communication with a wired and/or wireless network. Allsuch embodiments are envisioned. For example, resonator definitionsystem 102 can comprise a server device, a computing device, ageneral-purpose computer, a special-purpose computer, a quantumcomputing device (e.g., a quantum computer, a quantum processor, etc.),a tablet computing device, a handheld device, a server class computingmachine and/or database, a laptop computer, a notebook computer, adesktop computer, a cell phone, a smart phone, a consumer applianceand/or instrumentation, an industrial and/or commercial device, adigital assistant, a multimedia Internet enabled phone, a multimediaplayers, and/or another type of device.

In some embodiments, resonator definition system 102 can be coupled(e.g., communicatively, electrically, operatively, etc.) to one or moreexternal systems, sources, and/or devices (e.g., computing devices,communication devices, etc.) via a data cable (e.g., coaxial cable,High-Definition Multimedia Interface (HDMI), recommended standard (RS)232, Ethernet cable, etc.). In some embodiments, resonator definitionsystem 102 can be coupled (e.g., communicatively, electrically,operatively, etc.) to one or more external systems, sources, and/ordevices (e.g., computing devices, communication devices, etc.) via anetwork.

According to multiple embodiments, such a network can comprise wired andwireless networks, including, but not limited to, a cellular network, awide area network (WAN) (e.g., the Internet) or a local area network(LAN). For example, resonator definition system 102 can communicate withone or more external systems, sources, and/or devices, for instance,computing devices (and vice versa) using virtually any desired wired orwireless technology, including but not limited to: wireless fidelity(Wi-Fi), global system for mobile communications (GSM), universal mobiletelecommunications system (UMTS), worldwide interoperability formicrowave access (WiMAX), enhanced general packet radio service(enhanced GPRS), third generation partnership project (3GPP) long termevolution (LTE), third generation partnership project 2 (3GPP2) ultramobile broadband (UMB), high speed packet access (HSPA), Zigbee andother 802.XX wireless technologies and/or legacy telecommunicationtechnologies, BLUETOOTH®, Session Initiation Protocol (SIP), ZIGBEE®,RF4CE protocol, WirelessHART protocol, 6LoWPAN (IPv6 over Low powerWireless Area Networks), Z-Wave, an ANT, an ultra-wideband (UWB)standard protocol, and/or other proprietary and non-proprietarycommunication protocols. In such an example, resonator definition system102 can thus include hardware (e.g., a central processing unit (CPU), atransceiver, a decoder), software (e.g., a set of threads, a set ofprocesses, software in execution) or a combination of hardware andsoftware that facilitates communicating information between resonatordefinition system 102 and external systems, sources, and/or devices(e.g., computing devices, communication devices, etc.).

According to multiple embodiments, resonator definition system 102 cancomprise one or more computer and/or machine readable, writable, and/orexecutable components and/or instructions that, when executed byprocessor 106, can facilitate performance of operations defined by suchcomponent(s) and/or instruction(s). Further, in numerous embodiments,any component associated with resonator definition system 102, asdescribed herein with or without reference to the various figures of thesubject disclosure, can comprise one or more computer and/or machinereadable, writable, and/or executable components and/or instructionsthat, when executed by processor 106, can facilitate performance ofoperations defined by such component(s) and/or instruction(s). Forexample, resonant circuit component 108, resonator definition component110, and/or any other components associated with resonator definitionsystem 102 as disclosed herein (e.g., communicatively, electronically,and/or operatively coupled with and/or employed by resonator definitionsystem 102), can comprise such computer and/or machine readable,writable, and/or executable component(s) and/or instruction(s).Consequently, according to numerous embodiments, resonator definitionsystem 102 and/or any components associated therewith as disclosedherein, can employ processor 106 to execute such computer and/or machinereadable, writable, and/or executable component(s) and/or instruction(s)to facilitate performance of one or more operations described hereinwith reference to resonator definition system 102 and/or any suchcomponents associated therewith.

In some embodiments, resonator definition system 102 can facilitate(e.g., via processor 106) performance of operations executed by and/orassociated with resonant circuit component 108, resonator definitioncomponent 110, and/or another component associated with resonatordefinition system 102 as disclosed herein. For example, as described indetail below, resonator definition system 102 can facilitate: deriving aresonant circuit indicative of a superconducting resonator of asuperconducting circuit based on one or more attributes of thesuperconducting circuit; and/or defining a frequency value of thesuperconducting resonator based on the resonant circuit. In someembodiments, resonator definition system 102 can further facilitate:defining a length value of the superconducting resonator based on theresonant circuit; deriving the resonant circuit based on at least one ofa kinetic inductance value of a material of the superconducting circuitor a capacitive coupling value of a qubit capacitively coupled to atleast one of the resonant circuit, the superconducting resonator, or aground; deriving the resonant circuit based on a total capacitivecoupling value comprising a capacitive coupling value of the resonantcircuit and one or more capacitive coupling values of one or more qubitscapacitively coupled to at least one of the resonant circuit, thesuperconducting resonator, or a ground; deriving the resonant circuitbased on a total inductance value comprising an inductance value of theresonant circuit and one or more kinetic inductance values of one ormore materials of the superconducting circuit; and/or deriving theresonant circuit based on a defined frequency value of thesuperconducting resonator or a defined length value of thesuperconducting resonator. In some embodiments, resonator definitionsystem 102 can further facilitate: determining a kinetic inductancevalue of a superconducting circuit; and/or defining one or moreparameters of a superconducting resonator based on the kineticinductance value. In some embodiments, resonator definition system 102can further facilitate defining at least one of a frequency value of thesuperconducting resonator or a length value of the superconductingresonator based on at least one of the kinetic inductance value or acapacitive coupling value of a qubit capacitively coupled to at leastone of the superconducting circuit, the superconducting resonator, or aground.

According to multiple embodiments, resonant circuit component 108 canderive a resonant circuit indicative of a superconducting resonator of asuperconducting circuit based on one or more attributes of thesuperconducting circuit. For example, in some embodiments, resonantcircuit component 108 can derive a resonant circuit such as, forinstance, a quantum inductor capacitor circuit (also referred to hereinand/or denoted in the figures as an LC circuit, an LC model, a quantumLC circuit, a quantum LC model, a tank circuit, a tuned circuit, etc.).As referenced herein, a resonant circuit can comprise an inductor (L)coupled (e.g., electrically) to a capacitor (C) and a ground, where sucha resonant circuit can operate as an electrical resonator, storingenergy that oscillates at the resonant frequency of the resonantcircuit. For example, the capacitor (C) can store energy in an electricfield present between the plates of such capacitor (C) and the inductor(L) can store energy in a magnetic field surrounding such an inductor(L).

In some embodiments, resonant circuit component 108 can derive (e.g.,generate) a resonant circuit indicative of (e.g., representative ofand/or that can simulate) a superconducting resonator of asuperconducting circuit. For example, resonant circuit component 108 canderive a resonant circuit (e.g., an LC model, a quantum LC model, etc.)that can represent one or more parameters (e.g., frequency, length,etc.) of a superconducting resonator (e.g., a transmission line)implemented in a superconducting circuit (e.g., a circuit quantumelectrodynamics (circuit QED)). In another example, resonant circuitcomponent 108 can derive a resonant circuit (e.g., an LC model, aquantum LC model, etc.) that can simulate operation of a superconductingresonator (e.g., a transmission line) implemented in a superconductingcircuit (e.g., a circuit quantum electrodynamics (circuit QED)).

In some embodiments, resonator definition system 102 can generate anelectrical circuit that can constitute a model of a superconductingcircuit. For example, resonator definition system 102 can generatecircuit 300 a, circuit 300 b, and/or circuit 400 described below andillustrated in FIGS. 3 and 4, respectively, where such circuits cancomprise electrical circuits that can constitute models ofsuperconducting circuits.

FIGS. 3A and 3B illustrate example, non-limiting circuits 300 a, 300 bthat can facilitate superconducting resonator definition based on one ormore superconducting circuit attributes in accordance with one or moreembodiments described herein. Repetitive description of like elementsand/or processes employed in respective embodiments is omitted for sakeof brevity. In some embodiments, as described above, circuit 300 aand/or circuit 300 b can comprise electrical circuits that canconstitute models of superconducting circuits.

According to multiple embodiments, circuit 300 a (FIG. 3A) can comprisea superconducting circuit having a superconducting resonator that can bemodeled (e.g., by resonator definition system 102) as a transmissionline 302. In some embodiments, transmission line 302 can comprise alength value (l), a frequency value (f), and/or an impedance (Z₀) thatcan be defined by an entity (e.g., by a human using a graphical userinterface (GUI) of resonator definition system 102, not illustrated inthe figures). In some embodiments, transmission line 302 can becapacitively coupled to one or more qubits, where such capacitivecoupling of each qubit can be modeled (e.g., by resonator definitionsystem 102) as two capacitors, one in series (e.g., denoted as Cg₁ andCg₂ in FIGS. 3A and 3B) and one in parallel (e.g., denoted as Cp₁ andCp₂ in FIGS. 3A and 3B) between transmission line 302 and one or moregrounds 304 a, 304 b, as illustrated in FIGS. 3A and 3B. In someembodiments, such capacitive coupling of each qubit that can berepresented by capacitors Cg₁, Cg₂, Cp₁, and Cp₂ can comprise discretecapacitive coupling values respectively corresponding to capacitors Cg₁,Cg₂, Cp₁, and Cp₂, where such capacitive coupling values are alsodenoted herein as Cg₁, Cg₂, Cp₁, and Cp₂, respectively.

According to multiple embodiments, circuit 300 b (FIG. 3B) can comprisea superconducting circuit having a superconducting resonator that can bemodeled (e.g., by resonator definition system 102 and/or resonantcircuit component 108) as a resonant circuit 306. For example, resonantcircuit component 108 can derive resonant circuit 306, where resonantcircuit 306 can comprise a model (e.g., a representation) oftransmission line 302 described above and illustrated in FIG. 3A. Insome embodiments, resonant circuit 306 can comprise a capacitor (e.g.,denoted as C in FIG. 3B) and an inductor (e.g., denoted as L in FIG.3B). In some embodiments, capacitor C can comprise a capacitive couplingvalue corresponding thereto that is also denoted herein as C andinductor L can have an inductance value corresponding thereto that isalso denoted herein as L.

In some embodiments, resonant circuit component 108 can derive aresonant circuit based on a defined frequency value of a superconductingresonator. For example, by employing equations (1) and (2) definedbelow, resonant circuit component 108 can derive resonant circuit 306based on a frequency value f of a superconducting resonator (e.g.,transmission line 302) that can be defined by an entity (e.g., a humanusing a GUI of resonator definition system 102 as described above). Inanother example, by employing equation (3) defined below, resonantcircuit component 108 can calculate a length value l of asuperconducting resonator (e.g., transmission line 302) based on afrequency value f of such a superconducting resonator that can bedefined by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above).

$\begin{matrix}{C = \frac{1}{4fZ_{0}}} & (1) \\{L = \frac{1}{4\pi^{2}f^{2}C}} & (2) \\{l = \frac{c}{2f_{\sqrt{\epsilon_{r_{eff}}}}}} & (3)\end{matrix}$

where:

C denotes the capacitive coupling value of resonant circuit 306 (e.g.,the equivalent capacitance);

f denotes a frequency value of a superconducting resonator that cancomprise transmission line 302 and/or can be modeled (e.g., by resonatordefinition system 102 and/or resonant circuit component 108) as resonantcircuit 306;

Z₀ denotes a line impedance value (e.g., an impedance value oftransmission line 302);

c denotes the speed of light;

l denotes a length value of a superconducting resonator that cancomprise transmission line 302, and therefore, l can denote a lengthvalue of transmission line 302; and

ε_(r) _(eff) denotes an effective dielectric constant of a dielectricsubstrate that can be used to fabricate a superconducting resonatorcomprising transmission line 302.

Additionally, or alternatively, in some embodiments, resonant circuitcomponent 108 can derive a resonant circuit based on a defined lengthvalue of a superconducting resonator. For example, by employingequations (4), (5), and (6) defined below, resonant circuit component108 can derive resonant circuit 306 based on a length value l of asuperconducting resonator (e.g., transmission line 302) that can bedefined by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above).

$\begin{matrix}{f_{\lambda/2} = \frac{c}{2l\sqrt{\epsilon_{r_{eff}}}}} & (4) \\{C = \frac{1}{4f_{\lambda/2}Z_{0}}} & (5) \\{L = \frac{1}{4\pi^{2}f_{\lambda/2}^{2}C}} & (6)\end{matrix}$

where f_(λ/2) denotes a frequency value of a superconducting resonatorthat can comprise transmission line 302 and/or can be modeled (e.g., byresonator definition system 102 and/or resonant circuit component 108)as resonant circuit 306.

In some embodiments, resonant circuit 306 can be capacitively coupled toone or more qubits, where such capacitive coupling of each qubit can bemodeled (e.g., by resonator definition system 102) as two capacitors,one in series (e.g., denoted as Cg₁ and Cg₂ in FIGS. 3A and 3B) and onein parallel (e.g., denoted as Cp₁ and Cp₂ in FIGS. 3A and 3B) betweentransmission line 302 and one or more grounds 304 a, 304 b, asillustrated in FIGS. 3A and 3B. In some embodiments, resonant circuit306 can be further coupled (e.g., electrically) to a ground 308, asillustrated in FIG. 3B. In some embodiments, resonant circuit 306 can becapacitively coupled to a single qubit (e.g., which can be representedby capacitors Cg₁ and Cp₁ or capacitors Cg₂ and Cp₂) and a capacitor(not illustrated in the figures) to facilitate readout of such a qubit.

In some embodiments, resonant circuit component 108 can derive aresonant circuit indicative of (e.g., representative of and/or that cansimulate) a superconducting resonator including, but not limited to, asingle mode superconducting resonator, a single mode qubit busresonator, a single mode qubit transmission line resonator, a singlemode transmission line superconducting resonator, and/or anothersuperconducting resonator. For example, resonant circuit component 108can derive resonant circuit 306 and/or resonant circuit 402 describedbelow with reference to FIG. 4, where resonant circuit 306 and/orresonant circuit 402 can each be indicative of a single modesuperconducting resonator comprising a single mode (e.g., singlefrequency) near a qubit frequency. For instance, resonant circuit 306and/or resonant circuit 402 can each comprise a representation (e.g., amodel) of a single mode superconducting resonator that can operate at asingle frequency (e.g., can generate and/or identify a signal at acertain frequency) that is approximately equivalent to the frequency ofone or more qubits coupled to resonant circuit 306 and/or resonantcircuit 402.

Returning to FIG. 1, in some embodiments, resonant circuit component 108can derive a resonant circuit indicative of a superconducting resonatorof a superconducting circuit based on one or more attributes of thesuperconducting circuit, where such one or more attributes of thesuperconducting circuit can include, but are not limited to, a kineticinductance value of the superconducting circuit, a capacitive couplingvalue(s) of one or more qubits of the superconducting circuit, and/oranother attribute of the superconducting circuit. For example, resonantcircuit component 108 can derive a resonant circuit based on one or morekinetic inductance values of one or more materials (e.g., dielectricsubstrate(s), metal(s), superconducting metal(s), etc.) of asuperconducting circuit and/or one or more capacitive coupling values ofone or more qubits of such a superconducting circuit (e.g., capacitivecoupling value(s) of one or more qubits capacitively coupled to theresonant circuit, the superconducting resonator, and/or a ground).

As referenced herein, kinetic inductance can comprise an equivalentseries inductance representing the inertial mass of mobile chargecarriers in alternative electric fields, where such kinetic inductancecan comprise an additional inductive component of a superconductingresonator (e.g., a transmission line such as, for instance, transmissionline 302) that can change the frequency of such a superconductingresonator. In some embodiments, as such kinetic inductance of a material(e.g., a dielectric substrate, a superconducting metal, etc.) used tofabricate a superconducting resonator can change the frequency of such asuperconducting resonator, resonant circuit component 108 can derive aresonant circuit that can account for such a change in the frequency ofsuch a superconducting resonator. For example, as described below andillustrated in FIG. 4, resonant circuit component 108 can derive aresonant circuit (e.g., resonant circuit 402) that can account for sucha change in the frequency of such a superconducting resonator. Forinstance, resonant circuit component 108 can derive resonant circuit 402described below and illustrated in FIG. 4, where resonant circuitcomponent 108 can derive resonant circuit 402 based on a kineticinductance value of a material (e.g., a dielectric substrate, a metal,etc.) that can be used to fabricate transmission line 302 and/or one ormore capacitive coupling values of one or more qubits of circuit 300 aand/or circuit 300 b (e.g., capacitive coupling values of capacitors Cp₁and Cp₂ illustrated in FIGS. 3A and 3B).

FIG. 4 illustrates an example, non-limiting circuit 400 that canfacilitate superconducting resonator definition based on one or moresuperconducting circuit attributes in accordance with one or moreembodiments described herein. Repetitive description of like elementsand/or processes employed in respective embodiments is omitted for sakeof brevity. In some embodiments, as described above, circuit 400 cancomprise an electrical circuit that can constitute a model of asuperconducting circuit.

According to multiple embodiments, circuit 400 can comprise asuperconducting circuit having a superconducting resonator that can bemodeled (e.g., by resonator definition system 102 and/or resonantcircuit component 108) as a resonant circuit 402. In some embodiments,similar to resonant circuit 306 described above with reference to FIG.3B, resonant circuit component 108 can derive resonant circuit 402 basedon a defined frequency value of a superconducting resonator or a definedlength value of a superconducting resonator. For example, resonantcircuit component 108 can employ equations (1) and (2) defined above toderive resonant circuit 402 based on a frequency value f of asuperconducting resonator (e.g., transmission line 302) that can bedefined by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above). In another example, resonant circuitcomponent 108 can employ equations (4), (5), and (6) defined above toderive resonant circuit 402 based on a length value l of asuperconducting resonator (e.g., transmission line 302) that can bedefined by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above). Additionally, or alternatively, in someembodiments, as described above, resonant circuit component 108 canderive resonant circuit 402 based on a kinetic inductance value of amaterial (e.g., a dielectric substrate, a superconducting metal, etc.)that can be used to fabricate transmission line 302 and/or one or morecapacitive coupling values of one or more qubits of circuit 300 a and/orcircuit 300 b (e.g., capacitive coupling values of capacitors Cp₁ andCp₂ illustrated in FIGS. 3A and 3B).

In some embodiments, resonant circuit component 108 can derive resonantcircuit 402 based on a total capacitive coupling value of resonantcircuit 402 (e.g., denoted as C_(T) in FIG. 4) and/or a total inductancevalue of resonant circuit 402 (e.g., denoted as L_(T) in FIG. 4). Insome embodiments, such a total capacitive coupling value C_(T) ofresonant circuit 402 can comprise a capacitive coupling value ofcapacitor C of resonant circuit 306 (e.g., the equivalent capacitance)and capacitive coupling values of one or more capacitors Cp₁ and Cp₂ ofcircuit 300 a and/or circuit 300 b. In some embodiments, such totalinductance value L_(T) of resonant circuit 402 can comprise aninductance value of inductor L of resonant circuit 306 (e.g., theequivalent inductance) and a kinetic inductance value of a material(e.g., a dielectric substrate, a superconducting metal, etc.) used tofabricate, for example, transmission line 302.

In some embodiments, to derive resonant circuit 402 based on such atotal capacitive coupling value C_(T) of resonant circuit 402, resonantcircuit component 108 can employ equations (7) or (8) defined below tocalculate such a total capacitive coupling value C_(T) of resonantcircuit 402. In some embodiments, for example, where a length value l ofa superconducting resonator (e.g., transmission line 302) is defined byan entity (e.g., a human using a GUI of resonator definition system 102as described above), resonant circuit component 108 can employ equations(4) and (5) defined above and equation (7) defined below to calculatesuch a total capacitive coupling value C_(T) of resonant circuit 402. Insome embodiments, for example, where a frequency value f of asuperconducting resonator (e.g., transmission line 302) is defined by anentity (e.g., a human using a GUI of resonator definition system 102 asdescribed above), resonant circuit component 108 can employ equation (1)defined above and equation (8) defined below to calculate such a totalcapacitive coupling value C_(T) of resonant circuit 402.C _(T) =C+C _(p) ₁ +C _(p) ₂   (7)C _(T) =C−C _(p) ₁ −C _(p) ₂   (8)

where:

C_(T) denotes the total capacitive coupling value of resonant circuit402;

C denotes the capacitive coupling value of resonant circuit 306 (e.g.,the equivalent capacitance), which can be calculated (e.g., by resonantcircuit component 108) based on a defined length value l using equations(4) and (5) defined above or based on a defined frequency value f usingequation (1) defined above; and

C_(p1) and C_(p2) denote the capacitive coupling values of capacitorsC_(p1) and C_(p2) that can represent two (2) qubits capacitively coupledto resonant circuit 306, the superconducting resonator (e.g.,transmission line 302), and/or a ground (e.g., grounds 304 a, 304 billustrated in FIG. 3).

In some embodiments, such capacitive coupling values of capacitorsC_(p1) and C_(p2) can be predetermined and/or input to resonatordefinition system 102. For example, such capacitive coupling values ofcapacitors C_(p1) and C_(p2) can be predetermined by simulating suchqubits and/or superconducting circuit using a simulation component(e.g., a simulation application such as, for instance, electromagneticsimulation software, an electromagnetic field solver application, etc.)of, for instance, resonator definition system 102 (not illustrated inthe figures). In another example, such capacitive coupling values ofcapacitors C_(p1) and C_(p2) can be input to resonator definition system102 by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above).

In some embodiments, to derive resonant circuit 402 based on such atotal inductance value L_(T) of resonant circuit 402, resonant circuitcomponent 108 can employ equations (9) or (10) defined below tocalculate such a total inductance value L_(T) of resonant circuit 402.In some embodiments, for example, where a length value l of asuperconducting resonator (e.g., transmission line 302) is defined by anentity (e.g., a human using a GUI of resonator definition system 102 asdescribed above), resonant circuit component 108 can employ equations(4), (5), and (6) defined above and equation (9) defined below tocalculate such a total inductance value L_(T) of resonant circuit 402.In some embodiments, for example, where a frequency value f of asuperconducting resonator (e.g., transmission line 302) is defined by anentity (e.g., a human using a GUI of resonator definition system 102 asdescribed above), resonant circuit component 108 can employ equations(1) and (2) defined above and equation (10) defined below to calculatesuch a total inductance value L_(T) of resonant circuit 402.L _(T) =L+L _(kin)  (9)L _(T) =L−L _(kin)  (10)

where:

L_(T) denotes the total inductance value of resonant circuit 402;

L denotes the inductance value of resonant circuit 306 (e.g., theequivalent inductance), which can be calculated (e.g., by resonantcircuit component 108) based on a defined length value l using equations(4), (5), and (6) defined above or based on a defined frequency value fusing equations (1) and (2) defined above; and;

L_(kin) denotes the kinetic inductance value of a material (e.g., adielectric substrate, a metal, a superconducting metal, etc.) that canbe used to fabricate the superconducting resonator (e.g., transmissionline 302).

In some embodiments, to determine the kinetic inductance value L_(kin)defined above, resonant circuit component 108 can employ circuit effectcomponent 202. For example, resonant circuit component 108 can employcircuit effect component 202 to calculate such a kinetic inductancevalue L_(kin) using equation (11) as described below with reference toFIG. 2.

FIG. 2 illustrates a block diagram of an example, non-limiting system200 that can facilitate superconducting resonator definition based onone or more superconducting circuit attributes in accordance with one ormore embodiments described herein. Repetitive description of likeelements and/or processes employed in respective embodiments is omittedfor sake of brevity. In some embodiments, system 200 can compriseresonator definition system 102, which can comprise a circuit effectcomponent 202.

According to multiple embodiments, circuit effect component 202 candetermine a kinetic inductance value of a superconducting circuit. Forexample, circuit effect component 202 can determine the kineticinductance value L_(kin) defined above with reference to FIG. 4, wheresuch kinetic inductance value L_(kin) can comprise the kineticinductance value of a material (e.g., a dielectric substrate, a metal, asuperconducting metal, etc.) that can be used to fabricate transmissionline 302.

In some embodiments, circuit effect component 202 can calculate suchkinetic inductance value L_(kin) using equation (11) defined below.

$\begin{matrix}{L_{kin}^{\prime} = {\mu_{0}\lambda_{L}\frac{C}{4{{ADK}(k)}}\left( {\frac{1.7}{\sinh\left\lbrack {t/\left( {2\lambda_{L}} \right)} \right\rbrack} + \frac{0.4}{\sqrt{\left\lbrack {\left( {B/A} \right)^{2} - 1} \right\rbrack\left\lbrack {1 - \left( {B/D} \right)^{2}} \right\rbrack}}} \right)}} & (11)\end{matrix}$

where terms A, B, C, and D of equation (11) are defined below, and whereterm C does not denote capacitor C and thus, should not be confused withcapacitor C described above with reference to FIG. 3B:

${A = {{- \frac{t}{\pi}} + {\frac{1}{2}\sqrt{\left( \frac{2t}{\pi} \right)^{2} + \omega^{2}}}}},{B = \frac{\omega^{2}}{4A}},{C = {B - \frac{t}{\pi} + \sqrt{\left( \frac{t}{\pi} \right)^{2} + {\frac{1}{4}\left( {d - \omega} \right)^{2}}}}},{D = {\frac{2t}{\pi} + {C.}}}$

μ₀ denotes a magnetic permeability;

λ_(L) denotes a magnetic penetration depth;

K(k) denotes a complete elliptic integral of the first kind;

h denotes a height of a substrate (e.g., a dielectric substrate) uponwhich transmission line 302 can be formed;

t denotes a thickness of transmission line 302;

d denotes a spacing distance of transmission line 302;

w denotes a width of transmission line 302;

Returning to FIG. 1, according to multiple embodiments, resonatordefinition component 110 can define one or more parameters of asuperconducting resonator based on a resonant circuit. For example,resonator definition component 110 can define one or more parameters ofa superconducting resonator including, but not limited to, a frequencyvalue, a length value, and/or another parameter based on a resonantcircuit. For instance, resonator definition component 110 can definesuch one or more parameters of a superconducting resonator based onresonant circuit 306 and/or resonant circuit 402 described above withreference to FIGS. 3B and 4, where resonant circuit 306 and/or resonantcircuit 402 can comprise electrical circuit representations of such asuperconducting resonator (e.g., electrical circuit representations oftransmission line 302).

In some embodiments, resonator definition component 110 can define afrequency value of a superconducting resonator based on resonant circuit306, where resonant circuit 306 can comprise an electrical circuitrepresentation of such a superconducting resonator. For example, todefine such a frequency value of such a superconducting resonator basedon resonant circuit 306, resonator definition component 110 cancalculate a frequency value of resonant circuit 306 using equation (12)defined below.

$\begin{matrix}{f_{LC} = \frac{1}{2\pi\sqrt{LC}}} & (12)\end{matrix}$

where f_(LC) denotes a frequency value of resonant circuit 306 and asresonant circuit 306 can comprise an electrical circuit representationof a superconducting resonator, f_(LC) can therefore denote a frequencyvalue of such a superconducting resonator.

In some embodiments, the inductance value L and/or capacitive couplingvalue C of resonant circuit 306 can be calculated using one or more ofequations (1), (2), (4), (5), and/or (6) as described above withreference to FIG. 3B. For example, in some embodiments, resonatordefinition component 110 can employ resonant circuit component 108 tocalculate the inductance value L and/or capacitive coupling value C ofresonant circuit 306 using one or more of equations (1), (2), (4), (5),and/or (6) as described above with reference to FIG. 3B.

In some embodiments, resonator definition component 110 can define afrequency value of a superconducting resonator based on resonant circuit402, where resonant circuit 402 can comprise an electrical circuitrepresentation of such a superconducting resonator. For example, todefine such a frequency value of such a superconducting resonator basedon resonant circuit 402, resonator definition component 110 cancalculate a frequency value of resonant circuit 402 using equation (13)defined below.

$\begin{matrix}{f_{LC} = \frac{1}{2\pi\sqrt{L_{T}C_{T}}}} & (13)\end{matrix}$

where f_(LC) denotes a frequency value of resonant circuit 402 and asresonant circuit 402 can comprise an electrical circuit representationof a superconducting resonator, f_(LC) can therefore denote a frequencyvalue of such a superconducting resonator.

In some embodiments, the total inductance value L_(T) and/or totalcapacitive coupling value C_(T) of resonant circuit 402 can becalculated using one or more of equations (7), (8), (9), (10), and/or(11) as described above with reference to FIGS. 2 and 4. For example, insome embodiments, resonator definition component 110 can employ resonantcircuit component 108 to calculate the total inductance value L_(T)and/or total capacitive coupling value C_(T) of resonant circuit 402using one or more of equations (7), (8), (9), and/or (10) as describedabove with reference to FIG. 4. In this example, resonator definitioncomponent 110 can employ circuit effect component 202 to calculatekinetic inductance value L_(kin) using equation (11) as described abovewith reference to FIG. 2.

In some embodiments, resonant circuit component 108 can derive aresonant circuit based on one or more attributes of a superconductingcircuit comprising the resonant circuit and resonator definitioncomponent 110 can define one or more parameters of a superconductingresonator based on the resonant circuit. In these embodiments, resonatordefinition component 110 can therefore define such one or moreparameters of the superconducting resonator based on such one or moreattributes of the superconducting circuit. For example, resonatordefinition component 110 can define one or more parameters (e.g.,frequency value, length value, etc.) of a superconducting resonator(e.g., transmission line 302) based on one or more attributes of asuperconducting circuit (e.g., circuit 300 a, 300 b, 400, etc.)comprising the superconducting resonator such as, for instance, akinetic inductance value L_(kin) of a material (e.g., a dielectricsubstrate, a metal, a superconducting metal, etc.) that can be used tofabricate the superconducting resonator and/or one or more capacitivecoupling values of one or more qubits of the superconducting circuit(e.g., capacitive coupling values of capacitors C_(g1), C_(g2), C_(p1),and C_(p2) described above with reference to FIGS. 3A, 3B, and 4). Forinstance, as described above with reference to equation (13), resonatordefinition component 110 can define a frequency value of asuperconducting resonator based on resonant circuit 402, where resonantcircuit 402 can be derived by resonant circuit component 108 based onattributes of circuit 400 comprising a kinetic inductance value L_(kin)and capacitive coupling values of capacitors C_(p1) and C_(p2).

In some embodiments, resonator definition component 110 can define afrequency value of a superconducting resonator by applying an empiricalcorrection to the frequency value f_(LC) that can be calculated asdescribed above using equation (13), where such empirical correction canaccount for a change in the frequency value f_(LC) caused by thecapacitive coupling values of capacitors C_(g1) and C_(g2). For example,based on the frequency value f_(LC) calculated above using equation(13), resonator definition component 110 can calculate an empiricalcorrection value using equation (14) or (15) defined below.

In some embodiments, for example, where a length value l of asuperconducting resonator (e.g., transmission line 302) is defined by anentity (e.g., a human using a GUI of resonator definition system 102 asdescribed above), resonator definition component 110 can employequations (7), (9), and (13) defined above and equation (14) definedbelow to calculate such an empirical correction value. In someembodiments, for example, where a frequency value f of a superconductingresonator (e.g., transmission line 302) is defined by an entity (e.g., ahuman using a GUI of resonator definition system 102 as describedabove), resonator definition component 110 can employ equations (8),(10), and (13) defined above and equation (15) defined below tocalculate such an empirical correction value.Δf′ _(LC) ˜−Z ₀(C _(g1) +C _(g2))f _(LC) ²  (14)Δf′ _(LC) ˜Z ₀(C _(g1) +C _(g2))f _(LC) ²  (15)

where:

Δf′_(LC) denotes an empirical correction value comprising a change inthe frequency value f_(LC) caused by the capacitive coupling values ofcapacitors C_(g1) and C_(g2); and

C_(g1) and C_(g2) denote the capacitive coupling values of capacitorsC_(g1) and C_(g2) that can represent two (2) qubits capacitively coupledto resonant circuit 402.

In some embodiments, such capacitive coupling values of capacitorsC_(g1) and C_(g2) can be predetermined and/or input to resonatordefinition system 102. For example, such capacitive coupling values ofcapacitors C_(g1) and C_(g2) can be predetermined by simulating suchqubits and/or superconducting circuit using a simulation component(e.g., a simulation application such as, for instance, electromagneticsimulation software, an electromagnetic field solver application, etc.)of, for instance, resonator definition system 102 (not illustrated inthe figures). In another example, such capacitive coupling values ofcapacitors C_(g1) and C_(g2) can be input to resonator definition system102 by an entity (e.g., a human using a GUI of resonator definitionsystem 102 as described above).

In some embodiments, resonator definition component 110 can define afrequency value of a superconducting resonator based on the frequencyvalue f_(LC) calculated above using equation (13) and the empiricalcorrection value Δf′_(LC) calculated above using equation (14) or (15).For example, resonator definition component 110 can define a frequencyvalue of a superconducting resonator by mathematically combining (e.g.,summing, adding, etc.) the frequency value f_(LC) and the empiricalcorrection value Δf′_(LC).

In some embodiments, resonator definition component 110, can calculate acorrected length value of a superconducting resonator based on thefrequency value f_(LC) calculated above using equation (13) and theempirical correction value Δf′_(LC) calculated above using equation (14)or (15). For example, resonator definition component 110, can calculatea corrected length value of a superconducting resonator based on thefrequency value f_(LC) and the empirical correction value Δf′_(LC) usingequation (16) defined below.

$\begin{matrix}{l_{\lambda/2} = \frac{c}{2\left( {f_{LC} - {\Delta\; f_{LC}^{\prime}}} \right)\sqrt{\epsilon_{r_{eff}}}}} & (16)\end{matrix}$

where l_(λ/2) denotes a corrected length value of a superconductingresonator that accounts for the kinetic inductance value L_(kin), thecapacitive coupling values of C_(p1) and C_(p2), and the empiricalcorrection value Δf′_(LC) described above with reference to FIGS. 1, 2,3A, 3B, and 4.

FIG. 5 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 500 that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.Repetitive description of like elements and/or processes employed inrespective embodiments is omitted for sake of brevity.

In some embodiments, resonator definition system 102 (e.g., resonantcircuit component 108, resonator definition component 110, circuiteffect component 202, etc.) can execute computer-implemented method 500to define a frequency value of a superconducting resonator that accountsfor the kinetic inductance value L_(kin), the capacitive coupling valuesof C_(p1) and C_(p2), and the empirical correction value Δf′_(LC)described above with reference to FIGS. 1, 2, 3A, 3B, and 4. In someembodiments, computer-implemented method 500 can comprise an iterativecomputer-implemented method that can facilitate defining such afrequency value of a superconducting resonator based on a defined lengthvalue (e.g., an initial and/or assumed length value) of thesuperconducting resonator, where such a defined length value can beadjusted iteratively to obtain a desired frequency value (e.g., apredefined frequency value).

In some embodiments, at 502, computer-implemented method 500 cancomprise defining, by a system (e.g., via resonator definition system102, resonant circuit component 108, a GUI of resonator definitionsystem 102, etc.) operatively coupled to a processor (e.g., processor106), a length value of a superconducting resonator. For example, anentity (e.g., a human using a GUI of resonator definition system 102 asdescribed above) can input such a defined length value (e.g., an initialand/or assumed length value) of a superconducting resonator (e.g.,transmission line 302, a superconducting resonator represented byresonant circuit 306, a superconducting resonator represented byresonant circuit 402, etc.).

In some embodiments, at 504, computer-implemented method 500 cancomprise estimating (e.g., calculating), by the system (e.g., viaresonator definition system 102 and/or resonant circuit component 108),a frequency value of a superconducting resonator based on the definedlength value (e.g., a frequency value of a superconducting resonatorsuch as, for instance, transmission line 302, a superconductingresonator represented by resonant circuit 306, a superconductingresonator represented by resonant circuit 402, etc.). For example,resonator definition system 102 and/or resonant circuit component 108can estimate (e.g., calculate) such a frequency value (e.g., frequencyvalue f_(λ/2)) using equation (4) as described above with reference toFIGS. 3A and 3B.

In some embodiments, at 506, computer-implemented method 500 cancomprise deriving (e.g., generating), by the system (e.g., via resonatordefinition system 102 and/or resonant circuit component 108), an LCmodel (e.g., resonant circuit 306). For example, by employing equations(5) and (6) described above with reference to FIGS. 3A and 3B, resonantcircuit component 108 can derive (e.g., generate) an LC model (e.g., aquantum inductor capacitor circuit) such as, for instance, resonantcircuit 306 based on the frequency value (e.g., frequency value f_(λ/2))calculated at 504 described above.

In some embodiments, at 508, computer-implemented method 500 cancomprise receiving, by the system (e.g., via resonator definition system102, resonant circuit component 108, a GUI of resonator definitionsystem 102, etc.), qubit design data. For example, resonator definitionsystem 102 (e.g., via a GUI of resonator definition system 102) and/orresonant circuit component 108 can receive capacitive coupling values ofcapacitors C_(p1), C_(p2), C_(g1), and/or C_(g2) defined above, wheresuch capacitive coupling values of capacitors C_(p1), C_(p2), C_(g1),and/or C_(g2) can be predetermined by simulation of qubits representedby such capacitors and/or a superconducting circuit comprising suchqubits as described above.

In some embodiments, at 510, computer-implemented method 500 cancomprise modifying (e.g., updating), by the system (e.g., via resonatordefinition system 102, resonant circuit component 108, resonatordefinition component 110, and/or circuit effect component 202), the LCmodel. For example, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify resonant circuit 306 (e.g., a frequencyvalue of resonant circuit 306) to account for the kinetic inductancevalue L_(kin) defined above with reference to FIG. 4 and/or thecapacitive coupling values of capacitors C_(p1) and/or C_(p2). Inanother example, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify resonant circuit 306 and/or resonantcircuit 402 (e.g., a frequency value of resonant circuit 306 and/orresonant circuit 402) to account for the capacitive coupling values ofcapacitors C_(g1) and/or C_(g2).

In some embodiments, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify (e.g., update) resonant circuit 306(e.g., a frequency value of resonant circuit 306) to account for thekinetic inductance value L_(kin) by calculating the total inductancevalue L_(T) using equation (9) defined above and the kinetic inductancevalue L_(kin) using equation (11) defined above (e.g., using circuiteffect component 202). In these embodiments, resonator definition system102 (e.g., via resonant circuit component 108, resonator definitioncomponent 110, and circuit effect component 202) can further modifyresonant circuit 306 (e.g., a frequency value of resonant circuit 306)to account for the capacitive coupling values of capacitors C_(p1)and/or C_(p2) by calculating total capacitive coupling value C_(T) usingequation (7) defined above.

In these embodiments, by calculating the total inductance value L_(T)and the total capacitive coupling value C_(T), resonator definitionsystem 102 (e.g., via resonant circuit component 108, resonatordefinition component 110, and circuit effect component 202) can therebyderive resonant circuit 402, where resonant circuit 402 can comprise afrequency value f_(LC) that can be based on such total inductance valueL_(T) and total capacitive coupling value C_(T). In these embodiments,resonator definition system 102 (e.g., via resonant circuit component108, resonator definition component 110, and circuit effect component202) can calculate such a frequency value f_(LC) using equation (13)defined above. In these embodiments, resonator definition system 102(e.g., via resonant circuit component 108, resonator definitioncomponent 110, and circuit effect component 202) can modify (e.g.,update) resonant circuit 402 (e.g., modify the frequency value f_(LC) ofresonant circuit 402) to account for the capacitive coupling values ofcapacitors C_(g1) and/or C_(g2) by calculating the empirical correctionvalue Δ′f_(LC) using equation (14) defined above. In these embodiments,resonator definition system 102 (e.g., via resonant circuit component108, resonator definition component 110, and circuit effect component202) can modify frequency value f_(LC) of resonant circuit 402 toaccount for the empirical correction value Δf′_(LC) by mathematicallycombining (e.g., summing, adding, etc.) the empirical correction valueΔf′_(LC) and the frequency value f_(LC).

In some embodiments, at 512, computer-implemented method 500 cancomprise determining, by the system (e.g., via resonator definitionsystem 102, resonant circuit component 108, resonator definitioncomponent 110, circuit effect component 202, etc.), whether the modifiedfrequency value f_(LC) of resonant circuit 402 calculated at 510 aboveis acceptable. For example, resonator definition system 102 (e.g., viaresonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) can determine whether the modifiedfrequency value f_(LC) of resonant circuit 402 calculated at 510 aboveis within a defined numerical range (e.g., within a range ofapproximately 1 megahertz (MHz) to approximately 15 MHz) of a definedfrequency value of a qubit (e.g., 5 gigahertz (GHz), 10 GHz, etc.) thatcan be capacitively coupled to resonant circuit 402.

In some embodiments, if the system (e.g., resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines at 512 that the modifiedfrequency value f_(LC) of resonant circuit 402 is not acceptable, at514, computer-implemented method 500 can comprise adjusting, by thesystem (e.g., via resonator definition system 102, resonant circuitcomponent 108, a GUI of resonator definition system 102, etc.) thelength value of the superconducting resonator defined at 502 above. Inthese embodiments, based on the adjusted length value of thesuperconducting resonator adjusted at 514 above, the system (e.g.,resonator definition system 102, resonant circuit component 108,resonator definition component 110, circuit effect component 202, etc.)can repeat, as needed, operations 502 to 514 of computer-implementedmethod 500 to obtain a modified frequency value f_(LC) of resonantcircuit 402 that is acceptable (e.g., within a defined numerical rangeof a defined frequency of a qubit). In these embodiments, such anacceptable modified frequency value f_(LC) of resonant circuit 402 canthereby comprise a frequency value of a superconducting resonator thataccounts for the kinetic inductance value L_(kin), the capacitivecoupling values of C_(p1) and C_(p2), and the empirical correction valueΔf′_(LC) described above with reference to FIGS. 1, 2, 3A, 3B, and 4.

In some embodiments, if the system (e.g., resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines at 512 that the modifiedfrequency value f_(LC) of resonant circuit 402 is acceptable, at 516,computer-implemented method 500 can comprise ending, by the system(e.g., via resonator definition system 102, resonant circuit component108, resonator definition component 110, circuit effect component 202,etc.) computer-implemented method 500.

FIG. 6 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 600 that can facilitate a superconductingresonator definition process based on one or more superconductingcircuit attributes in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

In some embodiments, resonator definition system 102 (e.g., resonantcircuit component 108, resonator definition component 110, circuiteffect component 202, etc.) can execute computer-implemented method 600to define a length value of a superconducting resonator that accountsfor the kinetic inductance value L_(kin), the capacitive coupling valuesof C_(p1) and C_(p2), and the empirical correction value Δf′_(LC)described above with reference to FIGS. 1, 2, 3A, 3B, and 4. In someembodiments, computer-implemented method 600 can comprise an iterativecomputer-implemented method that can facilitate defining such a lengthvalue of a superconducting resonator based on a defined frequency value(e.g., an initial, assumed, and/or desired frequency value) of thesuperconducting resonator, where such a defined frequency value can beadjusted iteratively to obtain a desired length value (e.g., apredefined length value).

In some embodiments, at 602, computer-implemented method 600 cancomprise defining, by a system (e.g., via resonator definition system102, resonant circuit component 108, a GUI of resonator definitionsystem 102, etc.) operatively coupled to a processor (e.g., processor106), a frequency of a superconducting resonator. For example, an entity(e.g., a human using a GUI of resonator definition system 102 asdescribed above) can input such a defined frequency value (e.g., aninitial, assumed, and/or desired frequency value) of a superconductingresonator (e.g., transmission line 302, a superconducting resonatorrepresented by resonant circuit 306, a superconducting resonatorrepresented by resonant circuit 402, etc.).

In some embodiments, at 604, computer-implemented method 600 cancomprise estimating (e.g., calculating), by the system (e.g., viaresonator definition system 102 and/or resonant circuit component 108),a length value of a superconducting resonator based on the definedfrequency value (e.g., a length value of a superconducting resonatorsuch as, for instance, transmission line 302, a superconductingresonator represented by resonant circuit 306, a superconductingresonator represented by resonant circuit 402, etc.). For example,resonator definition system 102 and/or resonant circuit component 108can estimate (e.g., calculate) such a length value (e.g., length valuel) using equation (3) as described above with reference to FIGS. 3A and3B.

In some embodiments, at 606, computer-implemented method 600 cancomprise deriving (e.g., generating), by the system (e.g., via resonatordefinition system 102 and/or resonant circuit component 108), an LCmodel (e.g., resonant circuit 306). For example, by employing equations(1) and (2) described above with reference to FIGS. 3A and 3B, resonantcircuit component 108 can derive (e.g., generate) an LC model (e.g., aquantum inductor capacitor circuit) such as, for instance, resonantcircuit 306 based on the frequency value defined at 602 above and/or thelength value (e.g., length value l) calculated at 604 described above.

In some embodiments, at 608, computer-implemented method 600 cancomprise receiving, by the system (e.g., via resonator definition system102, resonant circuit component 108, a GUI of resonator definitionsystem 102, etc.), qubit design data. For example, resonator definitionsystem 102 (e.g., via a GUI of resonator definition system 102) and/orresonant circuit component 108 can receive capacitive coupling values ofcapacitors C_(p1), C_(p2), C_(g1), and/or C_(g2) defined above, wheresuch capacitive coupling values of capacitors C_(p1), C_(p2), C_(g1),and/or C_(g2) can be predetermined by simulation of qubits representedby such capacitors and/or a superconducting circuit comprising suchqubits as described above.

In some embodiments, at 610, computer-implemented method 600 cancomprise modifying (e.g., updating), by the system (e.g., via resonatordefinition system 102, resonant circuit component 108, resonatordefinition component 110, and/or circuit effect component 202), the LCmodel. For example, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify resonant circuit 306 (e.g., a frequencyvalue of resonant circuit 306) to account for the kinetic inductancevalue L_(kin) defined above with reference to FIG. 4 and/or thecapacitive coupling values of capacitors C_(p1) and/or C_(p2). Inanother example, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify resonant circuit 306 and/or resonantcircuit 402 (e.g., a frequency value of resonant circuit 306 and/orresonant circuit 402) to account for the capacitive coupling values ofcapacitors C_(g1) and/or C_(g2).

In some embodiments, resonator definition system 102 (e.g., via resonantcircuit component 108, resonator definition component 110, and circuiteffect component 202) can modify (e.g., update) resonant circuit 306(e.g., a frequency value of resonant circuit 306) to account for thekinetic inductance value L_(kin) by calculating the total inductancevalue L_(T) using equation (10) defined above and the kinetic inductancevalue L_(kin) using equation (11) defined above (e.g., using circuiteffect component 202). In these embodiments, resonator definition system102 (e.g., via resonant circuit component 108, resonator definitioncomponent 110, and circuit effect component 202) can further modifyresonant circuit 306 (e.g., a frequency value of resonant circuit 306)to account for the capacitive coupling values of capacitors C_(p1)and/or C_(p2) by calculating total capacitive coupling value C_(T) usingequation (8) defined above.

In these embodiments, by calculating the total inductance value L_(T)and the total capacitive coupling value C_(T), resonator definitionsystem 102 (e.g., via resonant circuit component 108, resonatordefinition component 110, and circuit effect component 202) can therebyderive resonant circuit 402, where resonant circuit 402 can comprise afrequency value f_(LC) that can be based on such total inductance valueL_(T) and total capacitive coupling value C_(T). In these embodiments,resonator definition system 102 (e.g., via resonant circuit component108, resonator definition component 110, and circuit effect component202) can calculate such a frequency value f_(LC) using equation (13)defined above. In these embodiments, resonator definition system 102(e.g., via resonant circuit component 108, resonator definitioncomponent 110, and circuit effect component 202) can modify (e.g.,update) resonant circuit 402 (e.g., modify the frequency value f_(LC) ofresonant circuit 402) to account for the capacitive coupling values ofcapacitors C_(g1) and/or C_(g2) by calculating the empirical correctionvalue Δf′_(LC) using equation (15) defined above. In these embodiments,resonator definition system 102 (e.g., via resonant circuit component108, resonator definition component 110, and circuit effect component202) can modify frequency value f_(LC) of resonant circuit 402 toaccount for the empirical correction value Δf′_(LC) by mathematicallycombining (e.g., summing, adding, etc.) the empirical correction valueΔf′_(LC) and the frequency value f_(LC).

In some embodiments, at 612, computer-implemented method 600 cancomprise determining, by the system (e.g., via resonator definitionsystem 102, resonant circuit component 108, resonator definitioncomponent 110, circuit effect component 202, etc.), whether a lengthvalue of a superconducting resonator represented by resonant circuit 402is acceptable, where such a length value can correspond to the modifiedfrequency value f_(LC) of resonant circuit 402 calculated at 610 above.For example, resonator definition system 102 (e.g., via resonant circuitcomponent 108, resonator definition component 110, circuit effectcomponent 202, etc.) can determine whether the modified frequency valuef_(LC) of resonant circuit 402 calculated at 610 above is within adefined numerical range (e.g., within a range of approximately 1 MHz toapproximately 15 MHz range) of a defined frequency value of a qubit(e.g., 5 GHz, 10 GHz, etc.) that can be capacitively coupled to resonantcircuit 402. In this example, if resonator definition system 102 (e.g.,via resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines the modified frequencyvalue f_(LC) of resonant circuit 402 calculated at 610 above is withinsuch a defined numerical range, resonator definition system 102 (e.g.,via resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) can thereby determine that thelength value of a superconducting resonator represented by resonantcircuit 402 is also acceptable.

In some embodiments, if the system (e.g., resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines at 612 that the modifiedfrequency value f_(LC) of resonant circuit 402 is not acceptable andtherefore the corresponding length value is also not acceptable, at 614,computer-implemented method 600 can comprise adjusting, by the system(e.g., via resonator definition system 102, resonant circuit component108, a GUI of resonator definition system 102, etc.) the frequency valueof the superconducting resonator defined at 602 above. In theseembodiments, based on the adjusted frequency value of thesuperconducting resonator adjusted at 614 above, the system (e.g.,resonator definition system 102, resonant circuit component 108,resonator definition component 110, circuit effect component 202, etc.)can repeat, as needed, operations 602 to 614 of computer-implementedmethod 600 to obtain a modified frequency value f_(LC) of resonantcircuit 402, and a corresponding length value, that is acceptable (e.g.,a modified frequency value f_(LC) that is within a defined numericalrange of a defined frequency of a qubit). In these embodiments, such anacceptable modified frequency value f_(LC) of resonant circuit 402, andcorresponding length value, can thereby comprise a frequency value and alength value of a superconducting resonator that account for the kineticinductance value L_(kin), the capacitive coupling values of C_(p1) andC_(p2), and the empirical correction value Δf′_(LC) described above withreference to FIGS. 1, 2, 3A, 3B, and 4.

In some embodiments, if the system (e.g., resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines at 612 that the modifiedfrequency value f_(LC) of resonant circuit 402 is not acceptable andtherefore the corresponding length value is also not acceptable,resonator definition system 102 (e.g., via resonant circuit component108, resonator definition component 110, circuit effect component 202,etc.), can calculate a corrected length value l_(λ/2) of asuperconducting resonator using equation (16) defined above. Forexample, using equation (16) defined above, resonator definition system102 (e.g., via resonant circuit component 108, resonator definitioncomponent 110, circuit effect component 202, etc.), can calculate acorrected length value l_(λ/2) of a superconducting resonator based onthe frequency value f_(LC) calculated above using equation (13) and theempirical correction value Δf′_(LC) calculated above using equation (14)or (15). In this example, resonator definition system 102 (e.g., viaresonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.), can calculate a corrected lengthvalue l_(λ/2) of a superconducting resonator that accounts for thekinetic inductance value L_(kin), the capacitive coupling values ofC_(p1) and C_(p2), and the empirical correction value Δf′_(LC) describedabove with reference to FIGS. 1, 2, 3A, 3B, and 4.

In some embodiments, if the system (e.g., resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) determines at 612 that the modifiedfrequency value f_(LC) of resonant circuit 402, and therefore thecorresponding length value, is acceptable, at 616, computer-implementedmethod 600 can comprise ending, by the system (e.g., via resonatordefinition system 102, resonant circuit component 108, resonatordefinition component 110, circuit effect component 202, etc.)computer-implemented method 600.

FIG. 7 illustrates an example, non-limiting table 700 that canfacilitate superconducting resonator definition based on one or moresuperconducting circuit attributes in accordance with one or moreembodiments described herein. Repetitive description of like elementsand/or processes employed in respective embodiments is omitted for sakeof brevity.

According to multiple embodiments, table 700 can comprise a column 702comprising superconducting resonator identification numbers (e.g., 1, 2,3, 4, etc.) of superconducting resonators that have been tested tomeasure frequency values of each superconducting resonator. In someembodiments, table 700 can comprise a column 704 comprising actual(e.g., real) measured frequency values (e.g., expressed in units ofgigahertz (GHz)) of fabricated superconducting resonators that have beenphysically measured.

In some embodiments, table 700 can comprise a column 706 comprisingfrequency error values (e.g., plus or minus (+/−) frequency valuesexpressed in units of megahertz (MHz)) of simulated superconductingresonators that have been measured using a simulation component (e.g., asimulation application such as, for instance, electromagnetic simulationsoftware, an electromagnetic field solver application, etc.) of, forinstance, resonator definition system 102 (not illustrated in thefigures). In these embodiments, such simulated superconductingresonators can be simulated based on the same parameters (e.g., length,material, etc.) used to fabricate the superconducting resonators ofcolumn 702. In some embodiments, column 706 can comprise frequency errorvalues indicative of a difference in frequency values of fabricatedsuperconducting resonators that have been physically measured (e.g.,measured frequency values of column 704) and frequency values ofsimulated superconducting resonators that have been simulated and/ormeasured based on the same parameters (e.g., length, material, etc.)used to fabricate the superconducting resonators of column 702.

In some embodiments, table 700 can comprise a column 708 comprisingfrequency error values (e.g., plus or minus (+/−) frequency valuesexpressed in units of megahertz (MHz)) of superconducting resonatorsmodeled as discrete resonant circuits (e.g., discrete resonant circuits402) that have been derived in accordance with one or more embodimentsdescribed herein, where each of such resonant circuits have a frequencyvalue corresponding thereto (e.g., the modified frequency value f_(LC)of resonant circuit 402 described above). In these embodiments, suchdiscrete resonant circuits (e.g., discrete resonant circuits 402) can bederived in accordance with one or more embodiments described hereinbased on the same parameters (e.g., length, material, etc.) used tofabricate the superconducting resonators of column 702. In someembodiments, column 708 can comprise frequency error values indicativeof a difference in frequency values of fabricated superconductingresonators that have been physically measured (e.g., measured frequencyvalues of column 704) and frequency values of resonant circuits (e.g.,modified frequency values fix of discrete resonant circuits 402) thathave been derived in accordance with one or more embodiments describedherein based on the same parameters (e.g., length, material, etc.) usedto fabricate the superconducting resonators of column 702.

In some embodiments, resonator definition system 102 can be associatedwith various technologies. For example, resonator definition system 102can be associated with classical computing technologies, quantumcomputing technologies, superconducting circuit technologies,superconducting circuit simulation technologies, inductor capacitorcircuit technologies, quantum inductor capacitor circuit technologies,resonator technologies, superconducting resonator technologies, and/orother technologies.

In some embodiments, resonator definition system 102 can providetechnical improvements to systems, devices, components, operationalsteps, and/or processing steps associated with the various technologiesidentified above. For example, resonator definition system 102 can:derive a resonant circuit indicative of a superconducting resonator of asuperconducting circuit based on one or more attributes of thesuperconducting circuit; and/or define a frequency value of thesuperconducting resonator based on the resonant circuit. In thisexample, resonator definition system 102 can thereby facilitate:improved accuracy of the frequency value (e.g., the modified frequencyvalue f_(LC) of resonant circuit 402 derived as described above); and/orimproved performance of the superconducting circuit due to such improvedaccuracy of the frequency value.

In this example, resonator definition system 102 can thereby reduce thetime and effort needed to design and/or simulate one or moresuperconducting resonators with proper accuracy, which can facilitateimproved performance of a processing unit (e.g., processor 106) byreducing processing time and/or processing workload of such a processingunit.

In some embodiments, resonator definition system 102 can providetechnical improvements to a processing unit (e.g., processor 106)associated with a classical computing device and/or a quantum computingdevice (e.g., a quantum processor, quantum hardware, superconductingcircuit, etc.). For example, by deriving a resonant circuit indicativeof a superconducting resonator of a superconducting circuit based on oneor more attributes of the superconducting circuit and/or defining afrequency value of the superconducting resonator based on the resonantcircuit, resonator definition system 102 can thereby reduce the time andeffort (e.g., human effort, computational effort and/or cost, etc.)needed to design and/or simulate one or more superconducting resonatorswith proper accuracy. In this example, by reducing the time and effortneeded to design and/or simulate one or more superconducting resonatorswith proper accuracy, resonator definition system 102 can therebyfacilitate improved performance of a processing unit (e.g., processor106) associated with resonator definition system 102 by reducingprocessing time and/or processing workload of such a processing unit.

In some embodiments, resonator definition system 102 can employ hardwareor software to solve problems that are highly technical in nature, thatare not abstract and that cannot be performed as a set of mental acts bya human In some embodiments, some of the processes described herein maybe performed by one or more specialized computers (e.g., one or morespecialized processing units, a specialized quantum computer, etc.) forcarrying out defined tasks related to the various technologiesidentified above. In some embodiments, resonator definition system 102and/or components thereof, can be employed to solve new problems thatarise through advancements in technologies mentioned above, employmentof quantum computing systems, cloud computing systems, computerarchitecture, and/or another technology.

It is to be appreciated that resonator definition system 102 can utilizevarious combinations of electrical components, mechanical components,and circuitry that cannot be replicated in the mind of a human orperformed by a human, as the various operations that can be executed byresonator definition system 102 and/or components thereof as describedherein are operations that are greater than the capability of a humanmind. For instance, the amount of data processed, the speed ofprocessing such data, or the types of data processed by resonatordefinition system 102 over a certain period of time can be greater,faster, or different than the amount, speed, or data type that can beprocessed by a human mind over the same period of time.

According to several embodiments, resonator definition system 102 canalso be fully operational towards performing one or more other functions(e.g., fully powered on, fully executed, etc.) while also performing thevarious operations described herein. It should be appreciated that suchsimultaneous multi-operational execution is beyond the capability of ahuman mind. It should also be appreciated that resonator definitionsystem 102 can include information that is impossible to obtain manuallyby an entity, such as a human user. For example, the type, amount, orvariety of information included in resonator definition system 102,resonant circuit component 108, resonator definition component 110,and/or circuit effect component 202 can be more complex than informationobtained manually by a human user.

FIG. 8 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 800 that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.Repetitive description of like elements and/or processes employed inrespective embodiments is omitted for sake of brevity.

In some embodiments, at 802, computer-implemented method 800 cancomprise deriving, by a system (e.g., via resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) operatively coupled to a processor(e.g., processor 106), a resonant circuit (e.g., resonant circuit 402)indicative of a superconducting resonator (e.g., a single modesuperconducting resonator modeled as resonant circuit 402) of asuperconducting circuit (e.g., circuit 400, a circuit QED, etc.) basedon one or more attributes of the superconducting circuit (e.g., akinetic inductance value of the superconducting circuit, capacitivecoupling value(s) of one or more qubits of the superconducting circuit).

In some embodiments, at 804, computer-implemented method 800 cancomprise defining, by the system (e.g., via resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) a frequency value (e.g., themodified frequency value f_(LC) of resonant circuit 402 described abovewith reference to FIG. 5) of the superconducting resonator based on theresonant circuit.

FIG. 9 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 900 that can facilitate superconductingresonator definition based on one or more superconducting circuitattributes in accordance with one or more embodiments described herein.Repetitive description of like elements and/or processes employed inrespective embodiments is omitted for sake of brevity.

In some embodiments, at 902, computer-implemented method 900 cancomprise determining, by a system (e.g., via resonator definition system102 and/or circuit effect component 202) operatively coupled to aprocessor (e.g., processor 106), a kinetic inductance value (e.g.,kinetic inductance value L_(kin) described above with reference to FIG.2) of a superconducting circuit (e.g., circuit 400, a circuit QED,etc.).

In some embodiments, at 904, computer-implemented method 900 cancomprise defining, by the system (e.g., via resonator definition system102, resonant circuit component 108, resonator definition component 110,circuit effect component 202, etc.) one or more parameters (e.g., themodified frequency value f_(LC) and/or the corrected length valuel_(λ/2) of resonant circuit 402 described above with reference to FIGS.5 and 6, respectively) of a superconducting resonator (e.g., a singlemode superconducting resonator modeled as resonant circuit 402) based onthe kinetic inductance value.

For simplicity of explanation, the computer-implemented methodologiesare depicted and described as a series of acts. It is to be understoodand appreciated that the subject innovation is not limited by the actsillustrated and/or by the order of acts, for example acts can occur invarious orders and/or concurrently, and with other acts not presentedand described herein. Furthermore, not all illustrated acts can berequired to implement the computer-implemented methodologies inaccordance with the disclosed subject matter. In addition, those skilledin the art will understand and appreciate that the computer-implementedmethodologies could alternatively be represented as a series ofinterrelated states via a state diagram or events. Additionally, itshould be further appreciated that the computer-implementedmethodologies disclosed hereinafter and throughout this specificationare capable of being stored on an article of manufacture to facilitatetransporting and transferring such computer-implemented methodologies tocomputers. The term article of manufacture, as used herein, is intendedto encompass a computer program accessible from any computer-readabledevice or storage media.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 10 as well as the following discussion are intendedto provide a general description of a suitable environment in which thevarious aspects of the disclosed subject matter can be implemented. FIG.10 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity.

With reference to FIG. 10, a suitable operating environment 1000 forimplementing various aspects of this disclosure can also include acomputer 1012. The computer 1012 can also include a processing unit1014, a system memory 1016, and a system bus 1018. The system bus 1018couples system components including, but not limited to, the systemmemory 1016 to the processing unit 1014. The processing unit 1014 can beany of various available processors. Dual microprocessors and othermultiprocessor architectures also can be employed as the processing unit1014. The system bus 1018 can be any of several types of busstructure(s) including the memory bus or memory controller, a peripheralbus or external bus, and/or a local bus using any variety of availablebus architectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and SmallComputer Systems Interface (SCSI).

The system memory 1016 can also include volatile memory 1020 andnonvolatile memory 1022. The basic input/output system (BIOS),containing the basic routines to transfer information between elementswithin the computer 1012, such as during start-up, is stored innonvolatile memory 1022. Computer 1012 can also includeremovable/non-removable, volatile/non-volatile computer storage media.FIG. 10 illustrates, for example, a disk storage 1024. Disk storage 1024can also include, but is not limited to, devices like a magnetic diskdrive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100drive, flash memory card, or memory stick. The disk storage 1024 alsocan include storage media separately or in combination with otherstorage media. To facilitate connection of the disk storage 1024 to thesystem bus 1018, a removable or non-removable interface is typicallyused, such as interface 1026. FIG. 10 also depicts software that acts asan intermediary between users and the basic computer resources describedin the suitable operating environment 1000. Such software can alsoinclude, for example, an operating system 1028. Operating system 1028,which can be stored on disk storage 1024, acts to control and allocateresources of the computer 1012.

System applications 1030 take advantage of the management of resourcesby operating system 1028 through program modules 1032 and program data1034, e.g., stored either in system memory 1016 or on disk storage 1024.It is to be appreciated that this disclosure can be implemented withvarious operating systems or combinations of operating systems. A userenters commands or information into the computer 1012 through inputdevice(s) 1036. Input devices 1036 include, but are not limited to, apointing device such as a mouse, trackball, stylus, touch pad, keyboard,microphone, joystick, game pad, satellite dish, scanner, TV tuner card,digital camera, digital video camera, web camera, and the like. Theseand other input devices connect to the processing unit 1014 through thesystem bus 1018 via interface port(s) 1038. Interface port(s) 1038include, for example, a serial port, a parallel port, a game port, and auniversal serial bus (USB). Output device(s) 1040 use some of the sametype of ports as input device(s) 1036. Thus, for example, a USB port canbe used to provide input to computer 1012, and to output informationfrom computer 1012 to an output device 1040. Output adapter 1042 isprovided to illustrate that there are some output devices 1040 likemonitors, speakers, and printers, among other output devices 1040, whichrequire special adapters. The output adapters 1042 include, by way ofillustration and not limitation, video and sound cards that provide ameans of connection between the output device 1040 and the system bus1018. It should be noted that other devices and/or systems of devicesprovide both input and output capabilities such as remote computer(s)1044.

Computer 1012 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)1044. The remote computer(s) 1044 can be a computer, a server, a router,a network PC, a workstation, a microprocessor based appliance, a peerdevice or other common network node and the like, and typically can alsoinclude many or all of the elements described relative to computer 1012.For purposes of brevity, only a memory storage device 1046 isillustrated with remote computer(s) 1044. Remote computer(s) 1044 islogically connected to computer 1012 through a network interface 1048and then physically connected via communication connection 1050. Networkinterface 1048 encompasses wire and/or wireless communication networkssuch as local-area networks (LAN), wide-area networks (WAN), cellularnetworks, etc. LAN technologies include Fiber Distributed Data Interface(FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ringand the like. WAN technologies include, but are not limited to,point-to-point links, circuit switching networks like IntegratedServices Digital Networks (ISDN) and variations thereon, packetswitching networks, and Digital Subscriber Lines (DSL). Communicationconnection(s) 1050 refers to the hardware/software employed to connectthe network interface 1048 to the system bus 1018. While communicationconnection 1050 is shown for illustrative clarity inside computer 1012,it can also be external to computer 1012. The hardware/software forconnection to the network interface 1048 can also include, for exemplarypurposes only, internal and external technologies such as, modemsincluding regular telephone grade modems, cable modems and DSL modems,ISDN adapters, and Ethernet cards.

The present invention may be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks and/or implement particular abstract datatypes. Moreover, those skilled in the art will appreciate that theinventive computer-implemented methods can be practiced with othercomputer system configurations, including single-processor ormultiprocessor computer systems, mini-computing devices, mainframecomputers, as well as computers, hand-held computing devices (e.g., PDA,phone), microprocessor-based or programmable consumer or industrialelectronics, and the like. The illustrated aspects can also be practicedin distributed computing environments in which tasks are performed byremote processing devices that are linked through a communicationsnetwork. However, some, if not all aspects of this disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

What is claimed is:
 1. A system, comprising: a memory that storescomputer executable components; and a processor that executes thecomputer executable components stored in the memory, wherein thecomputer executable components comprise: a resonator definitioncomponent that estimates a frequency value of a superconductingresonator of a superconducting circuit based on a length value of thesuperconducting resonator and an effective dielectric constant of adielectric substrate of the superconducting resonator; and a resonantcircuit component that generates a resonant circuit based on thefrequency value of the superconducting resonator, wherein the resonantcircuit models the superconducting resonator and comprises at least oneinductor and at least one capacitor.
 2. The system of claim 1, whereinthe resonant circuit comprises a quantum inductor capacitor circuit. 3.The system of claim 1, wherein the superconducting resonator comprises asingle mode superconducting resonator, a single mode qubit busresonator, a single mode qubit transmission line resonator, or a singlemode transmission line superconducting resonator.
 4. The system of claim1, wherein the resonant circuit component generates the resonant circuitbased further on one or more attributes of the superconducting circuit.5. The system of claim 4, wherein the one or more attributes comprise atleast one of a kinetic inductance value of a material of thesuperconducting circuit or a capacitive coupling value of a qubitcapacitively coupled to at least one of the resonant circuit, thesuperconducting resonator, or a ground, thereby facilitating at leastone of improved accuracy of the frequency value, improved performance ofthe superconducting circuit, or improved performance of the processor.6. The system of claim 4, wherein the one or more attributes comprise atotal capacitive coupling value comprising a capacitive coupling valueof the resonant circuit and one or more capacitive coupling values ofone or more qubits capacitively coupled to at least one of the resonantcircuit, the superconducting resonator, or a ground.
 7. The system ofclaim 4, wherein the one or more attributes comprise a total inductancevalue comprising an inductance value of the resonant circuit and one ormore kinetic inductance values of one or more materials of thesuperconducting circuit.
 8. A computer-implemented method, comprising:determining, by a system operatively coupled to a processor, a lengthvalue of a superconducting resonator of a superconducting circuit and aneffective dielectric constant of a dielectric substrate of thesuperconducting resonator; estimating, by the system, a frequency valueof the superconducting resonator based on the length value of thesuperconducting resonator and the effective dielectric constant of thedielectric substrate; and generating, by the system, a resonant circuitbased on the frequency value of the superconducting resonator, whereinthe resonant circuit models the superconducting resonator and comprisesat least one inductor and at least one capacitor.
 9. Thecomputer-implemented method of claim 8, wherein the resonant circuitcomprises a quantum inductor capacitor circuit.
 10. Thecomputer-implemented method of claim 8, wherein the superconductingresonator comprises a single mode superconducting resonator, a singlemode qubit bus resonator, a single mode qubit transmission lineresonator, or a single mode transmission line superconducting resonator.11. The computer-implemented method of claim 8, further comprisinggenerating, by the system, the resonant circuit based further on one ormore attributes of the superconducting circuit.
 12. Thecomputer-implemented method of claim 11, wherein the one or moreattributes comprise at least one of a kinetic inductance value of amaterial of the superconducting circuit or a capacitive coupling valueof a qubit capacitively coupled to at least one of the resonant circuit,the superconducting resonator, or a ground, thereby facilitating atleast one of improved accuracy of the frequency value, improvedperformance of the superconducting circuit, or improved performance ofthe processor.
 13. The computer-implemented method of claim 11, whereinthe one or more attributes comprise a total capacitive coupling valuecomprising a capacitive coupling value of the resonant circuit and oneor more capacitive coupling values of one or more qubits capacitivelycoupled to at least one of the resonant circuit, the superconductingresonator, or a ground.
 14. The computer-implemented method of claim 11,wherein the one or more attributes comprise a total inductance valuecomprising an inductance value of the resonant circuit and one or morekinetic inductance values of one or more materials of thesuperconducting circuit.
 15. A computer program product facilitatingsuperconducting resonator definition based on one or moresuperconducting circuit attributes, the computer program productcomprising a computer readable storage medium having programinstructions embodied therewith, the program instructions executable bya processor to cause the processor to: determine a length value of asuperconducting resonator of a superconducting circuit and an effectivedielectric constant of a dielectric substrate of the superconductingresonator; estimate a frequency value of the superconducting resonatorbased on the length value of the superconducting resonator and theeffective dielectric constant of the dielectric substrate; and generatea resonant circuit based on the frequency value of the superconductingresonator, wherein the resonant circuit models the superconductingresonator and comprises at least one inductor and at least onecapacitor.
 16. The computer program product of claim 15, wherein thesuperconducting resonator comprises a single mode superconductingresonator, a single mode qubit bus resonator, a single mode qubittransmission line resonator, or a single mode transmission linesuperconducting resonator.
 17. The computer program product of claim 15,wherein the program instructions are further executable by the processorto cause the processor to: generate the resonant circuit based furtheron one or more attributes of the superconducting circuit.
 18. Thecomputer program product of claim 17, wherein the one or more attributescomprise at least one of a kinetic inductance value of a material of thesuperconducting circuit or a capacitive coupling value of a qubitcapacitively coupled to at least one of the resonant circuit, thesuperconducting resonator, or a ground, thereby facilitating at leastone of improved accuracy of the frequency value, improved performance ofthe superconducting circuit, or improved performance of the processor.19. The computer program product of claim 17, wherein the one or moreattributes comprise a total capacitive coupling value comprising acapacitive coupling value of the resonant circuit and one or morecapacitive coupling values of one or more qubits capacitively coupled toat least one of the resonant circuit, the superconducting resonator, ora ground.
 20. The computer program product of claim 17, wherein the oneor more attributes comprise a total inductance value comprising aninductance value of the resonant circuit and one or more kineticinductance values of one or more materials of the superconductingcircuit.